### Video Transcript

What is the relation between the
focal length of a spherical mirror and the radius of curvature of the mirror? (A) The focal length of a spherical
mirror is double the radius of curvature of the mirror. (B) The focal length of a spherical
mirror is equal to the radius of curvature of the mirror. Or (C) the focal length of a
spherical mirror is half of the radius of curvature of the mirror.

In this question, we are being
asked about the relationship between the focal length of a spherical mirror and its
radius of curvature. Let’s start by drawing a ray
diagram for a spherical mirror.

We can draw our spherical mirror as
part of a circle, and we’ll add in the optical axis like this. We’ve said that this mirror has the
same shape as part of a circle. If we were to continue this so that
we had a full circle, the center of the circle would be here. This point is called the mirror’s
center of curvature. The mirror’s radius of curvature is
the distance between this point and the mirror, which we can add to the diagram like
this.

We can also mark the focal point of
the mirror. The focal point is a point which
all rays of light reflected by the mirror will pass through. It sits on the optical axis on the
same side as the center of curvature for a concave mirror. The focal length of the mirror is
the distance between this point and the mirror, measured along the optical axis. We know then that the focal length
of the mirror is this distance and that the radius of curvature is this
distance. We can see that the focal length is
shorter than the radius of curvature. In fact, for a concave spherical
mirror, the focal length is exactly half of the radius of curvature.

Looking through the answer options,
we can see this corresponds to option (C). The focal length of a spherical
mirror is half of the radius of curvature of the mirror. So, option (C) is the correct
answer.